Normality of Pi

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Write the unique nonterminating decimal expansion of $\pi$ as $\pi=3+\sum_{j\geq1}d_j10^{-j}$, with digits $d_j\in\{0,1,\ldots,9\}$. For a block $w=(w_1,\ldots,w_k)$ of length $k\geq1$, overlapping occurrences are counted by \[ A_w(N)=\#\{1\leq j\leq N:(d_j,d_{j+1},\ldots,d_{j+k-1})=w\}. \] Normality concerns every finite block, not merely equal frequencies of single digits.

Statement recorded in the supplied source.

For every $k\geq1$ and every block $w\in\{0,1,\ldots,9\}^k$, is \[ \lim_{N\to\infty}\frac{A_w(N)}N=10^{-k}? \] Equivalently, is $\pi$ normal in base $10$? [S2]

Short English statement

Does every finite string of decimal digits occur in the expansion of pi with its expected uniform limiting frequency?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1258 (NT-051). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Jeffrey C. Lagarias, On the Normality of Arithmetical Constants, arXiv:math/0101055v2 (2002), Introduction and Section 2, Definition 2.2. https://arxiv.org/pdf/math/0101055.